where s(n] = R n] denotes the state vector and A = the state transition matrix for our dynamic system model. The selection of the parameters a, b, c, d results in different dynamic scenarios. The fate of Romeo and Juliet's relationship depends on these model parameters (i.e. a, b, c, d) in the state transition matrix and the initial state (s[0]). In this problem, we'll explore some of these possibilities. (a) Consider the case where a + b = c + d in the state-transition matrix A = 2 a Show that is an eigenvector of A, and determine its corresponding eigenvalue 21. Show that V 2 is an eigenvector of A, and determine its corresponding eigenvalue 12. Now, express the first and second eigenvalues and their eigenspaces in terms of the parameters a, b, c, and d. Hint: You could use the characteristic polynomial approach to find the eigenvalues and eigenvectors. You may find it easier to use the following approach instead: . First find 21 by showing v1 = is an eigenvector of A. . Then find 12 by showing v2 = is an eigenvector of A. For parts (b) - (d), consider the following state-transition matrix: A = [0.75 0.25 0.25 0.75 (b) Determine the eigenpairs (i.e. (21, VI ) and (22, v2)) for this system. Note that this matrix is a special case of the matrix explored in part (a), so you can use results from that part to help you. (c) Determine all of the steady states of the system. That is, find the set of points such that if Romeo and Juliet start at, or enter, any of those points, their states will stay in place forever: {s, | As, = 5,}. (d) Suppose Romeo and Juliet start from an initial state $10] E span What happens to their relationship over time? Specifically, what is sin] as n - co? Now suppose we have the following state-transition matrix: Use this state-transition matrix for parts (e) - (g)